Asymptotic solution of the Schrödinger equation for the three body problem

Abstract

The W.K.B. method is applied to determine the asymptotic expansion of the Schrödinger wave function for a system of three spinless particles which interact through central forces. A new assumption is made concerning the form of the asymptotic wave function. This is designed to take advantage of the fact that the total angular momentum of the system is constant. As a result, new equations for the phase and amplitude of the wave function, different from those which follow from ordinary W.K.B. theory, are obtained. In particular, the phase function satisfies a Hamilton-Jacobi-like equation of degree 2(2L + 1). The method is applied to the helium atom for S- and P- states. The wave functions and energy eigenvalues are determined by means of perturbation theory. A classical description of the various electronic motions in the helium atom is presented. Previous attempts to describe the helium atom in terms of the old Bohr theory are clarified.